# minimum of different independent Poisson random variables

Let $X_1,\ldots,X_N$ be independent Poisson distributed random variables with unequal parameters $\lambda_1,\ldots,\lambda_N$.

Is there any closed form expression or at least a good approximation for the distribution (I am most interested in the CCDF) of their minimum $Y = \min\limits_{1\leqslant i\leqslant N}(X_{i})$?

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## 2 Answers

The CCDF of $Y$ is the product of the CCDF's of $X_1,\ldots, X_N$. The CCDF of $X_j$ (at nonnegative integer $x$) is $1 - \Gamma(1+x,\lambda_j)/x!$ where $\Gamma$ is the incomplete Gamma function. That's about as closed a form as you're going to get. As for approximations, which limit are you interested in?

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Thanks, Robert! I had the gamma formula as well. What I need to do is integrate this on $\lambda \in (0,\infty)$. Are there any good and integrable lower and/or upper bounds? – miladydesummer Dec 13 '12 at 21:14
Integrate what exactly and over what domain? If you the $\Lambda$'s a reasonably large then Poisson can be approximated by Normal distribution, so if you were interested in, say, the expectation of the minimum it will be bounded between $\min_i(\Lambda_i)$ and $\min_i(\lambda_i-\sqrt{\log(N)}\sqrt{\Lambda_i})$. – Ori Gurel-Gurevich Dec 14 '12 at 5:39
Oops, I accidentally made some of the $\lambda$'s $\Lambda$. – Ori Gurel-Gurevich Dec 14 '12 at 5:40
Sorry, my integration explanation was very vague. I want to integrate the CCDF of the minimum Y, over variable $t\in(0,\infty)$, with $\lambda_j = \p_j t$. Unfortunately, the $p_j$'s are very small, so I'm guessing the normal approximation would not be so great here... – miladydesummer Dec 14 '12 at 11:56
Do you really mean to integrate from $t=0$ to $\infty$? The CCDF goes to $1$ as $t \to \infty$, so the result would be $\infty$. – Robert Israel Dec 14 '12 at 18:43

For large $N$ asymptotics, you want to look into extreme value theory. In particular, take a look at this book.

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Isn't extreme value theory only for min/max of IID variables? Mine are independent but not identical, since they have different parameters... – miladydesummer Dec 15 '12 at 12:32
The most classical parts are but that's not the full extent of the theory, just like the most classical versions of the central limit theorem are for IID variables, but more general versions exist. – Mark Meckes Dec 17 '12 at 16:33